Bernard Nienhuis

• University of Amsterdam

Cherednik’s type A quantum affine Knizhnik–Zamolodchikov (qKZ) equations form a consistent system of linear q-difference equations for \(V_n\)-valued meromorphic functions on a complex n-torus, with \(V_n\) a module over the \(\mathrm{GL}_n\)-type extended affine Hecke algebra \({\mathcal {H}}_n\). The family \(({\mathcal {H}}_n)_{n\ge 0}\) of exte...

Cherednik's type A quantum affine Knizhnik-Zamolodchikov (qKZ) equations form a consistent system of linear $q$-difference equations for $V_n$-valued meromorphic functions on a complex $n$-torus, with $V_n$ a module over the $\mathrm{GL}_n$-type extended affine Hecke algebra $\mathcal{H}_n$. The family $(\mathcal{H}_n)_{n\geq 0}$ of extended affine...

Loop models are statistical ensembles of closed paths on a lattice. The most well-known among them has a variety of names such as the dense O(n) loop model, the Temperley-Lieb (TL) model. This note concerns the model in which the weight of the loop n = 1, and a local operator which changes the weight of all the loops that surround the position of t...

Molecular motor proteins fulfill the critical function of transporting organelles and other building blocks along the biopolymer network of the cell’s cytoskeleton, but crowding effects are believed to crucially affect this motor-driven transport due to motor interactions. Physical transport models, like the paradigmatic, totally asymmetric simple...

Using the totally asymmetric simple-exclusion-process and mean-field transport theory, we investigate the transport in closed random networks with simple crossing topology-two incoming, two outgoing segments, as a model for molecular motor motion along biopolymer networks. Inspired by in vitro observation of molecular motor motion, we model the mot...

In the framework of an inhomogeneous solvable lattice model, we derive exact expressions to boundary-to-boundary current on a lattice of finite width. The model we use is the dilute $O(n=1)$ loop model, and our expression are derived based on solutions of the $q$-Kniznik-Zamolodchikov equations, and recursion relations.

We introduce a new integrable supersymmetric lattice chain which violates fermion conservation and exhibits fermion-hole symmetry. The model displays exponential degeneracy in every eigenstate including the groundstate. This degeneracy is expressed in the possibility to create any number of zero modes reminiscent of Cooper pairs.

The transport of organelles and proteins is of vital importance for living cells. Besides passive transport by diffusion, active transport by molecular motors hopping over the cytoskeleton network is crucial for the survival of cells. (Experimental analysis) We use in vitro motility assays to take Totally Internal Reflection Microscopy (TIRF) imag...

The transport of organelles and proteins is of vital importance for living cells. Besides passive transport by diffusion, active transport by molecular motors hopping over the cytoskeleton network is crucial for the survival of cells. We performed simulations using the Totally Assymetric Exlusion Process (TASEP), a paradigmatic model for nonequilib...

The transport of organelles and proteins is of vital importance to living cells. For these large cellular components, diffusion is too slow and active transport along the cytoskeleton, driven by molecular motors is essential. We use in vitro motility assays to take Totally Internal Reflection Microscopy (TIRF) images of the movement of molecular mo...

This is the second part of our study of the ground state eigenvector of the transfer matrix of the dilute Temperley-Lieb loop model with the loop weight $n=1$ on a semi infinite strip of width $L$. We focus here on the computation of the normalization (otherwise called the sum rule) $Z_L$ of the ground state eigenvector, which is also the partition...

We consider the integrable dilute Temperley-Lieb (dTL) O($n=1$) loop model on a semi-infinite strip of finite width $L$. In the analogy with the Temperley-Lieb (TL) O($n=1$) loop model the ground state eigenvector of the transfer matrix is studied by means of a set of $q$-difference equations, sometimes called the $q$KZ equations. We compute some g...

We demonstrate that application of an increasing shear field on a glass leads to an intriguing dynamic first-order transition in analogy with equilibrium transitions. By following the particle dynamics as a function of the driving field in a colloidal glass, we identify a critical shear rate upon which the diffusion time scale of the glass exhibits...

The transport of organelles and proteins is of vital importance for living cells. Besides passive transport by diffusion, active transport by molecular motors hopping over the cytoskeleton network is crucial for the survival of cells. We performed simulations of this complex system using the Totally Asymmetric Exclusion Process (TASEP), to model th...

The transport of organelles and proteins is of vital importance for living cells. Besides passive transport by diffusion, active transport by molecular motors hopping over the cytoskeleton is crucial for the survival of cells. We study in vitro the movement of molecular motors over microtubule network. With Totally Internal Reflection Microscopy (T...

We demonstrate that application of an increasing shear field on a glass leads to an intriguing dynamic first order transition in analogy to equilibrium transitions. By following the particle dynamics as a function of the driving field in a colloidal glass, we identify a critical shear rate upon which the diffusion time scale of the glass exhibits a...

Via event-driven molecular dynamics simulations and experiments, we study the packing-fraction and shear-rate dependence of single-particle fluctuations and dynamic correlations in hard-sphere glasses under shear. At packing fractions above the glass transition, correlations increase as shear rate decreases: the exponential tail in the distribution...

The nature of the glass transition is one of the most important unsolved problems in condensed matter physics. The difference between glasses and liquids is believed to be caused by very large free energy barriers for particle rearrangements; however, so far it has not been possible to confirm this experimentally. We provide the first quantitative de...

Dynamic arrest is a general phenomenon across a wide range of dynamic systems including glasses, traffic flow, and dynamics in cells, but the universality of dynamic arrest phenomena remains unclear. We connect the emergence of traffic jams in a simple traffic flow model directly to the dynamic slowing down in kinetically constrained models for gla...

Spatial correlations of microscopic fluctuations are investigated via real-space experiments and computer simulations of colloidal glasses under steady shear. It is shown that while the distribution of one-particle fluctuations is always isotropic regardless of the relative importance of shear as compared to thermal fluctuations, their spatial corr...

Glasses behave as solids on experimental time scales due to their slow relaxation. Growing dynamic length scales due to cooperative motion of particles are believed to be central to this slow response. For quiescent glasses, however, the size of the cooperatively rearranging regions has never been observed to exceed a few particle diameters, and th...

We investigate a hard-square lattice gas on the square lattice by means of transfer-matrix and Monte Carlo methods. The size of the hard squares is equal to two lattice constants, so the simultaneous occupation of nearest-neighbor sites as well as of next-to-nearest-neighbor sites is excluded. Near saturation of the particle density, this system is...

Employing an inhomogeneous solvable lattice model, we derive an exact expression for a boundary-to-boundary current on a lattice of finite width. This current is an example of a class of parafermionic observables recently introduced in an attempt to rigorously prove conformal invariance of the scaling limit of critical two-dimensional lattice model...

Employing an inhomogeneous solvable lattice model, we derive an exact expression for a boundary-to-boundary edge current on a lattice of finite width. This current is an example of a class of parafermionic observables recently introduced in an attempt to rigorously prove conformal invariance of the scaling limit of critical two-dimensional lattice...

Distribution and correlations of contact forces in two-dimensional static granular packings is studied within a framework of (Edwards) Force Network Ensemble. A two-dimensional ``snooker'' model without friction is shown to be critical. An explicit expression for the force correlation function is derived using Bethe Ansatz. The force correlation fu...

Shear banding, i.e. the localization of shear flow, occurs in a manifold of systems ranging from hard materials such as metallic glasses to soft materials such as clays, shaving cream or mayonnaise. We investigate this phenomenon in a dense colloidal system using confocal microscopy that enables to track individual particles in 3D space and time. T...

We define a percolation problem on the basis of spin configurations of the two-dimensional XY model. Neighboring spins belong to the same percolation cluster if their orientations differ less than a certain threshold called the conducting angle. The percolation properties of this model are studied by means of Monte Carlo simulations and a finite-si...

We consider the Rényi alpha entropies for Luttinger liquids (LL). For large block lengths l, these are known to grow like lnl. We show that there are subleading terms that oscillate with frequency 2k{F} (the Fermi wave number of the LL) and exhibit a universal power-law decay with l. The new critical exponent is equal to K/(2alpha), where K is the...

We study two types of generalized Baxter-Wu models, by means of transfer-matrix and Monte Carlo techniques. The first generalization allows for different couplings in the up- and down triangles, and the second generalization is to a $q$-state spin model with three-spin interactions. Both generalizations lead to self-dual models, so that the probabl...

Great progress in the understanding of conformally invariant scaling limits of stochastic models, has been given by the Stochastic Löwner Evolutions (SLE). This approach has been pioneered by Schramm [46] and by Lawler, Schramm and Werner [31]. It describes a one-parameter family of conformally invariant measures of curves in the plane or a two-dim...

We show that the exactly solved low-temperature branch of the two-dimensional O(n) model is equivalent to an O(n) model with vacancies and a different value of n . We present analytic results for several universal parameters of the latter model, which is identified as a tricritical point. These results apply to the range n</=32 and include the exac...

We carry out a systematic study of the exact block entanglement in the XXZ spin chain at Δ = −1/2. We present the first analytic expressions for reduced density matrices for n spins in a chain of length L (for n≤6 and arbitrary but odd L) for a truly interacting model. The entanglement entropy and the moments of the reduced density matrix and its s...

Scaling properties of patterns formed by large contact forces are studied as a function of the applied shear stress, in two-dimensional static packings generated from the force network ensemble. An anisotropic finite-size-scaling analysis shows that the applied shear does not affect the universal scaling properties of these patterns, but simply ind...

We present exact results for several universal parameters of the tricritical O(n) model in two dimensions. The results apply to the range −2⩽n⩽3/2, and include the central charge and three scaling dimensions, associated with temperature, magnetic field and the introduction of an interface. Since these results are based on an extrapolation of known...

We study a special case of the Brauer model in which every path of the model has weight q = 1. The model has been studied before as a solvable lattice model and can be viewed as a Lorentz lattice gas. The paths of the model are also called self-avoiding trails. We consider the model in a triangle with boundary conditions such that one of the trails...

Force networks form the skeleton of static granular matter. They are the key factor that determines mechanical properties such as stability, elasticity and sound transmission, which are important for civil engineering and industrial processing. Previous studies have focused on investigations of the global structure of external forces (the boundary...

We propose exact expressions for the conformal anomaly and for three critical exponents of the tricritical O(n) loop model as a function of n in the range -2<or=n<or=3/2. These findings are based on an analogy with known relations between Potts and O(n) models and on an exact solution of a "tri-tricritical" Potts model described in the literature....

We study the geometry of forces in some simple models for granular stackings. The information contained in geometry is complementary to that in the distribution of forces in a single inter-particle contact, which is more widely studied. We present a method which focuses on the fractal nature of the force network and find good evidence of scale inva...

Starting from the Bethe ansatz solution we derive a set of coupled non-linear integral equations for the fully packed double loop model (FPL^2) on the square lattice. As an application we find exact expressions for the central charge and for the scaling dimension corresponding to the simplest charge excitation. We study numerically the low-lying ex...

We observe that the degree of the commuting variety and other related varieties occur as coefficients in the leading eigenvector of an integrable loop model based on the Brauer algebra.

We present conjectured exact expressions for two types of correlations in the dense O$(n=1)$ loop model on $L\times \infty$ square lattices with periodic boundary conditions. These are the probability that a point is surrounded by $m$ loops and the probability that $k$ consecutive points on a row are on the same or on different loops. The dense O$(...

We analyze a lattice model closely related to the one-dimensional inelastic gas with periodic boundary condition. The one-dimensional inelastic gas tends to form high density clusters of particles with almost the same velocity, separated by regions of low density; plotted as a function of particle indices, the velocities of the gas particles exhibi...

We determine the backbone exponent X(b) of several critical and tricritical q-state Potts models in two dimensions. The critical systems include the bond percolation, the Ising, the q=2-sqrt[3], 3, and 4 state Potts, and the Baxter-Wu model, and the tricritical ones include the q=1 Potts model and the Blume-Capel model. For this purpose, we formula...

We investigate geometric properties of the general q-state Potts model in two dimensions, and define geometric clusters as sets of lattice sites in the same Potts state, connected by nearest-neighbor bonds with variable probability p. We find that, besides the random-cluster fixed point, both the critical and the tricritical Potts models have anoth...

This article is meant to serve as a guide to recent developments in the study of the scaling limit of critical models. These new developments were made possible through the definition of the Stochastic Loewner Evolution (SLE) by Oded Schramm. This article opens with a discussion of Loewner's method, explaining how this method can be used to describ...

Conjectures for analytical expressions for correlations in the dense O$(1)$ loop model on semi infinite square lattices are given. We have obtained these results for four types of boundary conditions. Periodic and reflecting boundary conditions have been considered before. We give many new conjectures for these two cases and review some of the exis...

International audience We consider random paths on a square lattice which take a left or a right turn at every vertex. The possible turns are taken with equal probability, except at a vertex which has been visited before. In such case the vertex is left via the unused edge. When the initial edge is reached the path is considered completed. We also...

We decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons. We end up with a 24 vertex model which is known in the literature as the fully packed double loop model. In the particular case in which the fugacities of the...

In this note, we solve the Loewner equation in the upper half-plane with forcing function xi(t), for the cases in which xi(t) has a power-law dependence on time with powers 0, 1/2 and 1. In the first case the trace of singularities is a line perpendicular to the real axis. In the second case the trace of singularities can do three things. If xi(t)=...

We investigate a family of lattice models with manifest N=2 supersymmetry. The models describe fermions on a 1D lattice, subject to the constraint that no more than k consecutive lattice sites may be occupied. We discuss the special properties arising from the supersymmetry, and present Bethe ansatz solutions of the simplest models. We display the...

We discuss a one-dimensional model of a fluctuating interface with a dynamic exponent z=1. The events that occur are adsorption, which is local, and desorption which is nonlocal and may take place over regions of the order of the system size. In the thermodynamic limit, the time dependence of the system is given by characters of the c=0 logarithmic...

We propose a one-dimensional nonlocal stochastic model of adsorption and desorption depending on one parameter, the adsorption rate. At a special value of this parameter, the model has some interesting features. For example, the spectrum is given by conformal field theory, and the stationary non-equilibrium probability distribution is given by the...

1 Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3010, Australia 2 School of Mathematical Sciences, Australian National University, Canberra ACT 0200, Australia 3 Universiteit van Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands

A number of conjectures have been given recently concerning the connection between the antiferromagnetic XXZ spin chain at Δ = −1/2 and various symmetry classes of alternating sign matrices. Here we use the integrability of the XXZ chain to gain further insight into these developments. In doing so we obtain a number of new results using Baxter’s Q...

We examine the groundstate wavefunction of the rotor model for different boundary conditions. Three conjectures are made on the appearance of numbers enumerating alternating sign matrices. In addition to those occurring in the O(n = 1) model we find the number AV(2m + 1;3), which 3-enumerates vertically symmetric alternating sign matrices.

We examine the groundstate wavefunction of the rotor model for different boundary conditions. Three conjectures are made on the appearance of numbers enumerating alternating sign matrices. In addition to those occurring in the O(n = 1) model we find the number AV(2m + 1;3), which 3-enumerates vertically symmetric alternating sign matrices.

Recently, a model consisting of triangular trimers covering the triangular lattice was introduced and its exact free energy given. In this paper we present the complete calculation leading to this exact result. The solution involves a coordinate Bethe ansatz with two kinds of particles. It is similar to that of the square-triangle random tiling mod...

We consider the groundstate wavefunction of the quantum symmetric antiferromagnetic XXZ chain with open and twisted boundary conditions at $\Delta=-{1/2}$, along with the groundstate wavefunction of the corresponding O($n$) loop model at $n=1$. Based on exact results for finite-size systems, sums involving the wavefunction components, and in some c...

We establish exact results for coupled spin-1/2 chains for special values of the four-spin interaction V and dimerization parameter delta. The first exact result is at delta = 1/2 and V = -2. Because we find a very small but finite gap in this dimerized chain, this can serve as a very strong test case for numerical and approximate analytical techni...

We consider the ground-state wavefunction of the quantum symmetric antiferromagnetic XXZ chain with open and twisted boundary conditions at [iopmath latex="$\Delta=-\frac{1}{2}$"] = -½ [/iopmath] , along with the ground-state wavefunction of the corresponding [iopmath latex="$\Or(n)$"] O(n) [/iopmath] loop model at [iopmath latex="$n=1$"] n = 1 [/i...

Tiling models are classical statistical models in which different geometric shapes, the tiles, are packed together such that they cover space completely. In this paper we discuss a class of two-dimensional tiling models in which the tiles are rectangles and isosceles triangles. Some of these models have been solved recently by means of Bethe Ansatz...

We establish exact results for the one-dimensional spin-orbital model for special values of the four-spin interaction $V$ and dimerization parameter $\delta$. The first exact result is at $\delta=1/2$ and $V=-2$. Because we find a very small but finite gap in this dimerized chain, this can serve as a very strong test case for numerical and approxim...

Using Monte Carlo simulation, Van Duijneveldt and Lekkerkerker [Phys. Rev. Lett. 71, 4264 (1993)] found gas-liquid-solid behavior in a simple two-dimensional lattice model with two types of hard particles. The same model is studied here by means of numerical transfer-matrix calculations, focusing on the finite-size scaling of the gaps between the l...

The exact stationary state of an asymmetric exclusion process with fully parallel dynamics is obtained using the matrix product ansatz. We give a simple derivation for the deterministic case by a physical interpretation of the dimension of the matrices. We prove the stationarity via a cancellation mechanism, and by making use of an explicit represe...

A model is presented consisting of triangular trimers on the triangular lattice. In analogy to the dimer problem, these particles cover the lattice completely without overlap. The model has a honeycomb structure of hexagonal cells separated by rigid domain walls. The transfer matrix can be diagonalised by a Bethe Ansatz with two types of particles....

The phase diagram of the O(n) model, in particular the special case $n=0$, is studied by means of transfer-matrix calculations on the loop representation of the O(n) model. The model is defined on the square lattice; the loops are allowed to collide at the lattice vertices, but not to intersect. The loop model contains three variable parameters tha...

We explore the phase diagram of an O(n) model on the honeycomb lattice with vacancies, using finite-size scaling and transfer-matrix methods. We make use of the loop representation of the O(n) model, so that $n$ is not restricted to positive integers. For low activities of the vacancies, we observe critical points of the known universality class. A...

The authors describe the construction of regular lattices in two-dimensional hyperbolic space by means of the action of a discrete subgroups of SU(1,1). They consider an Ising model on such lattices and show how the thermodynamic limit can be handled. They give high- and low-temperature expansions of the free energy, magnetic susceptibility and mag...

A two-dimensional q-state Potts model with vacancies and four-spin interactions is studied. The parameter space of the model contains a critical and a tricritical manifold. Moreover for 0 less-than-or-equal-to q less-than-or-equal-to 9/4 a multicritical point is found which is the locus where the tricritical transition changes from first- to second...

The central charge of the Izergin-Korepin model the corresponding quantum spin chain, and the O(n) model is calculated analytically via the Bethe ansatz. The calculation extends a technique recently developed for the Zamolodchikov-Fateev model. In addition critical exponents and the central charge for these models are obtained from numerical soluti...

A cellular automaton which describes diffusion of particles with exclusion in one dimension is shown to be equivalent to a six-vertex model on a critical line. The arrow-arrow correlation function of the six-vertex model is calculated exactly on this line using a transfer matrix method.

We investigate the critical behaviour of the fully packed O(n) loop model on the square lattice in which each vertex is visited once by a loop. A transfer-matrix analysis shows that this model can be interpreted as a superposition of a low-temperature O(n) model and a solid-on-solid (SOS) model, as for the fully packed model on the honeycomb lattic...

An anisotropic statistical model on a cubic lattice consisting of locally interacting six-vertex planes solvable via Bethe ansatz (BA), is studied. Symmetries of BA lead to an infinite hierarchy of possible phases, which is further restricted by numerical simulations. The model is solved for an arbitrary value of the interlayer coupling constant. R...

Finite-size scaling and transfer-matrix techniques are used to determine the conformal anomaly and critical exponents of O(n) models on the square lattice. These calculations were performed on five branches of critical points parametrised by n. The results for two of the branches agree with the known universal properties of the O(n) model as derive...

A Potts model on a square lattice with two- and four-spin interaction and site and bond dilution is shown to be dual to itself. The model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model. By means of these mappings the dilute Potts model can be written as a Gaussian-like model with staggered and direct periodic...

The results of a variational renormalisation-group calculation for the magnetic exponent yH of the two-dimensional q-state Potts model suggest a simple relationship between yH and the exactly known critical exponent yT8v of the eight-vertex model. The relation allows one to predict the critical and tricritical magnetic exponent delta of the q-state...

To find critical points of O(n) models on the triangular lattice we apply two methods. First we investigate the Yang-Baxter equations on the triangular lattice. We find only solvable points directly related to those for the square lattice. Second we construct intersections with the Potts model. This yields eight branches of critical points, paramet...

The exact solution of the CSOS models by Pearce and Seaton is analysed. This solution appears to describe a first-order transition without droplet singularities. The authors show that the system with free boundaries has no such transition, whereas the system with fixed boundary conditions has a first-order transition at an anisotropy dependent locu...

A solid-on-solid (SOS) model in a field h conjugate to the orientation of the surface is exactly solved with the aid of Pfaffians. The free energy (h) directly gives the equilibrium shape of a finite crystal. The phase diagram exhibits rough and smooth phases, corresponding to rounded and flat portions of the crystal surface. The solid-on-solid mod...

Motivated by the study of motion in a random environment we introduce and investigate a variant of the Temperley - Lieb algebra. This algebra is very rich, providing us with three classes of solutions of the Yang - Baxter equation. This allows us to establish a theoretical framework to study the diffusive behaviour of a Lorentz lattice gas. Exact r...

The quasicrystalline state of matter and the role of quasiperiodicity is discussed. Both energetic and entropic mechanisms may stabilize the quasicrystalline phase. For systems where entropy plays the dominant role, random tiling models are the appropriate description. These are discrete statistical models, but without an underlying lattice. Severa...

Introduction Consider coverings of the triangular lattice with triangular trimers, such that each lattice site is occupied and the trimers do not overlap. In other words, each site should belong to exactly one trimer. Figure 1 shows a typical configuration. Figure 1: A typical configuration of the model. The triangular faces of the lattice are divi...

Introduction Consider a two-dimensional statistical mechanical model and put it on an infinite cylinder of circumference L. In the thermodynamic limit L ! 1 the free energy per unit length F L approaches f 1 L, where f 1 is the bulk free energy per unit area. For isotropic critical twodimensional systems conformal invariance predicts [1, 2] F L = f...

Introduction The phase behaviour of hard particles has received much attention. One of the questions is whether hard sphere mixtures can have more than one disordered phase. Motivated by this interest, Van Duijneveldt and Lekkerkerker [1, 2] studied a two-dimensional lattice model introduced by Frenkel and Louis [3]. It consists of hard hexagons an...

A formalism is presented to calculate the energy correlation functions of the two-dimensional q-state Potts model at its critical and tricritical transition, based on the Coulomb gas approach. The consistency of the procedure is verified by comparing alternative prescriptions for the calculation. The four-point function is calculated explicitly. Fr...

We study the scaling limits of the L-state Restricted Solid-on-Solid (RSOS) lattice models and their fusion hierarchies in the off-critical regimes. Starting with the elliptic functional equations of Klümper and Pearce, we derive the Thermodynamic Bethe Ansatz (TBA) equations of Zamolodchikov. Although this systematic approach, in principle, allows...

We numerically investigate the influence of self-attraction on the critical behaviour of a polymer in two dimensions, by means of an analysis of finite-size results of transfer-matrix calculations. The transfer matrix is constructed on the basis of the O(n) loop model in the limit n→0. It yields finite-size results for the magnetic correlation leng...

52C23 Quasicrystals, aperiodic tilings 60K35 Interacting random processes; statistical mechanics type models; percolation theory (See also 82B43, 82C43) 82B23 Exactly solvable models; Bethe ansatz

In this paper we investigate an integrable loop model and its connection with a supersymmetric spin chain. The Bethe Ansatz solution allows us to study some properties of the ground state. When the loop fugacity $q$ lies in the physical regime, we conjecture that the central charge is $c=q-1$ for $q$ integer $< 2$. Low-lying excitations are examine...

We develop a microscopic scattering theory for the electromagnetic response of dielectrics. We derive the Lorentz-Lorenz relation for a hard-sphere fluid and for hard-sphere mixtures by summing rigorously the relevant class of multiple scattering events which incorporates particle correlations. The derivation neither makes use of macroscopic concep...

We show that a rectangle triangle random tiling with a tenfold symmetric phase is solvable by Bethe Ansatz. After the twelvefold square triangle and the eightfold rectangle triangle random tiling, this is the third example of a rectangle triangle tiling which is solvable. A Bethe Ansatz solution provides in principle an accurate estimate of the ent...

Solvable via Bethe Ansatz (BA) anisotropic statistical model on cubic lattice consisting of locally interacting 6-vertex planes, is studied. Symmetries of BA lead to infinite hierarchy of possible phases, which is further restricted by numerical simulations. The model is solved for arbitrary value of the interlayer coupling constant. Resulting is t...